The price of gravel is $24 for every (3)/(8) ton. Kate wants to know the price of 2 tons of gravel.
The correct answer and explanation is :
To find the price of 2 tons of gravel when the price is $24 for every ( \frac{3}{8} ) ton, we can use the concept of unit rates and proportion. The key is to scale up the price for a full 2 tons based on the price for ( \frac{3}{8} ) ton.
Step-by-step process:
- Understand the given information:
The price of gravel is $24 for every ( \frac{3}{8} ) ton. This means that for every ( \frac{3}{8} ) of a ton, the cost is $24. - Find the price per ton:
To find the price for 1 ton of gravel, we need to set up a proportion. If $24 is the price for ( \frac{3}{8} ) ton, we want to know how much 1 ton costs. Set up the ratio: [
\frac{24}{\frac{3}{8}} = \text{Price for 1 ton}
] When dividing by a fraction, we multiply by its reciprocal, so we can rewrite the equation as: [
\frac{24}{\frac{3}{8}} = 24 \times \frac{8}{3}
] - Simplify:
[
24 \times \frac{8}{3} = \frac{24 \times 8}{3} = \frac{192}{3} = 64
] Therefore, the price for 1 ton of gravel is $64. - Find the price for 2 tons:
Since the price for 1 ton is $64, the price for 2 tons is simply twice that amount: [
2 \times 64 = 128
] Therefore, the price for 2 tons of gravel is $128.
Explanation:
In this problem, the key concept is to determine the cost per unit (per ton) and then multiply that by the desired quantity (2 tons). We first converted the given price for ( \frac{3}{8} ) ton into the cost per full ton, then scaled that to find the total cost for 2 tons.
This is an example of a rate problem, where we’re dealing with proportional relationships. By setting up a proportion and solving for the price per ton, we ensured that our calculation was based on the correct unit rate. The method of multiplying by the reciprocal to simplify the division by a fraction is an essential skill when working with rates.